Bivariate interpolatory rational splines

Smoothing conditions in terms of Bézier coefficients of piecewise rational functions on an arbitrary triangulation are derived. This facilitates the solution of the problem of bivariate rational spline interpolation, with or without convexity constraints, particularly on the three and four-directional meshes. For such a triangulation, we also derive the conformality condition that a bivariate rationale spline function must satisfy, and we demonstrate the interpolation scheme with a low-degree example.The research of this author was supported by NSF Grant # DMS-92-06928.     

Interpolation by integro quintic splines

Recently, Behforooz [1], has introduced a new approach to construct cubic splines by using the integral values, rather than the usual function values at the knots. Also he has established different sets of end conditions for cubic and quintic splines by using the integral values, see Behforooz [2], [3] and [4]. In this paper, we will use the same techniques of [1] to construct integro quintic splines. Although by using the integral values we expected to face a more complicated process for our construction, it turned out that the matrix of the system of linear equations that produces the parameters became a diagonally dominant matrix and the process became very simple. The selection of the required end conditions for our integro quintic splines will be discussed. The numerical examples and computational results illustrate and guarantee a higher accuracy for this approximation.     

Computation of interpolatory splines via triadic subdivision

We present an algorithm for the computation of interpolatory splines of arbitrary order at triadic rational points. The algorithm is based on triadic subdivision of splines. Explicit expressions for the subdivision symbols are established. These are rational functions. The computations are implemented by recursive filtering.     

Asymptotic analysis of boundary conditions for quintic splines

In this article, we consider various boundary conditions for interpolation of quintic splines of defect 1 on a uniform mesh. We obtain an asymptotic representation of the approximation error for the spline for different boundary conditions. Boundary conditions that are optimal by approximation accuracy are found.     

Improved orders of approximation derived from interpolatory cubic splines

Lets be a cubic spline, with equally spaced knots on [a, b] interpolating a given functiony at the knots. The parameters which determines are used to construct a piecewise defined polynomialP of degree four. It is shown thatP can be used to give better orders of approximation toy and its derivatives than those obtained froms. It is also shown that the known superconvergence properties of the derivatives ofs, at specific points of [a, b], are all special cases of the main result contained in the present paper.     

A best approximation property of discrete quadratic interpolatory splines

We investigate the following problem in this paper: where there is an unique 1-periodic discrete quadratic spline s∈S(3,p,h) satisfying certain interpolatory condition for a 1-periodic discrete function defined on [0,1]_h. The anwser is affirmative.     

End conditions for interpolatory cubic splines with unequally spaced knots

A class of end conditions is derived for cubic spline interpolation at unequally spaced knots. These conditions are in terms of function values at the knots and lead to 0 (h⁴) convergence uniformly on the interval of interpolation.     

An algorithm for the interpolation of functions using quintic splines

A method is described for the interpolation of N arbitrarily given data points using fifth degree polynomial spline functions. The interpolating spline is built from a set of basis functions belonging to the fifth degree smooth Hermite space. The resulting algebraic system is symmetric and bloc-tridiagonal. Its solution is calculated using a direct inversion method, namely a block-gaussian elimination without pivoting. Various boundary conditions are provided for independently at each end point. The stability of the algorithm is examined and some examples are given of experimental convergence rates for the interpolation of elementary analytical functions. A listing is given of the two FORTRAN subroutines INSPL5 and SPLIN5 which form the algorithm.     

Shape-preserving interpolation by G1 and G2 PH quintic splines

The interpolation of a planar sequence of points p₀, ..., p_Nby shape-preserving G¹ or G² PH quintic splines with specifiedend conditions is considered. The shape-preservation propertyis secured by adjusting ‘tension’ parameters thatarise upon relaxing parametric continuity to geometric continuity.In the G² case, the PH spline construction is based on applyingNewton–Raphson iterations to a global system of equations,commencing with a suitable initialization strategy—thisgeneralizes the construction described previously in NumericalAlgorithms 27, 35–60 (2001). As a simpler and cheaperalternative, a shape-preserving G¹ PH quintic spline schemeis also introduced. Although the order of continuity is lower,this has the advantage of allowing construction through purelylocal equations.     

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C ¹ smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots—that are not considered by the authors of the corresponding spline basis—thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data.     

A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines

This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes.     

Regularizing properties of explicit approximating splines

A method is proposed for the solution of some operator equations of the first kind by successive smoothing of the righthand side using an explicit spline approximation and by numerical inversion of the operator. For the relevant class of problems, the regularizing properties of the method are reduced on the whole to the regularizing properties of explicit spline approximation, which are studied under certain assumptions about errors in input data and the discretization step.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 39–46, 1986.     

Algebraic properties of discrete box splines

The central objective of this paper is to discuss linear independence of translates of discrete box splines which we introduced earlier as a device for the fast computation of multivariate splines. The results obtained here allow us to draw conclusions about the structure of such discrete splines which have, in particular, applications to counting the number of nonnegative integer solutions of linear diophantine equations.